Hamilton-Jacobi Equations on Networks as Limits of Singularly Perturbed Problems in Optimal Control: Dimension Reduction

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We consider a family of star-shaped planar domains Ωϵ, made of N non intersecting semi-infinite strips of thickness ϵ and of a central region whose diameter is proportional to ϵ. As ϵ → 0, Ωϵ tends to a network 𝒢 made of half-lines sharing an endpoint O. We study infinite horizon optimal control problems in which the state is constrained to remain in . We prove that the value function tends to the solution of a Hamilton-Jacobi equation on 𝒢, with an effective transmission condition at O. The effective equation is linked to an optimal control problem.